Question

In a class, $40$% of the students are members of the eco club and $25$% of the students are members of the mathematics club. If a student is selected at random from the class, what is the probability that the student is not a member of either of the clubs.

Answer 1

Let $n\left(s\right)=100$ and $A$ be the event of selecting a member of eco club.
$P\left(A\right)=\frac{40}{100}=\frac{2}{5}$
Let $B$ be the event of selecting a member of mathematics club.
$P\left(B\right)=\frac{25}{100}=\frac{1}{4}$
Since, $A$ and $B$ are mutually exclusive events
we have, $P(A\cup B)=P\left(A\right)+P\left(B\right)=\frac{2}{5}+\frac{1}{4}=\frac{13}{20}$
$\therefore$ the probability of selecting a student who is a member of eco club or mathematics club is $\frac{13}{20}$.
Now, the event $C$ of selecting a student who is not a member of either of the clubs is the complementary event of $(A\cup B)$.
Thus, $P(A\cup B)+P\left(C\right)=1$
$P\left(C\right)=1-P(A\cup B)=1-\frac{13}{20}=\frac{7}{20}$
$\therefore$ The probability of selecting a student who is not a member of either of the clubs is $\frac{7}{20}$